You have a choice when dealing with domain name services (DNS). Is the new host you are moving to also the domain name registrar? Meaning wherever you bought your domain name? If no, then you need to log into your domain name registrar account (like GoDaddy, NameCheap or wherever) and find the place in your account where you can tell the registrar where to point requests for your domain name. Those are the Name Servers listings. If you host at your registrar (e.g., GoDaddy) they will usually have a simple "hosted here" option (may not be those exact words, of course ;) ).
Guessing your new webhost is NOT where you registered your domain name. In that case you will want to log into your new hosting account and find their name server information. This is "usually" something along the lines of "NS1.NewWebHostCompany.com", "NS2.NewWebHostCompany.com", etc.
You will then take that NS info and insert it into your domain name account at your registrar.
Wait a while (some say 24 to 72 hours. Yikes!) and your new setting should be propagated throughout the global DNS system and when visitors type in (or click a link) to your sites, they will end up at the new host, not the old one.
Trust that is clear enough and helps out a bit.